3 Answers
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To augment STW's answer, a Maillard reaction is a browning chemical reaction between an amino acid and a sugar. Heat encourages these reactions. Carmelization is an example of a Maillard reaction, although there are others.

These reactions are responsible for the flavor of the crust of bread, as well as most nutty, caramel, and toffee notes in beer. These flavors are the result of compounds known as melanoidins, one of the byproducts of Maillard reactions.

In certain styles, such as bocks, dunkelweizens, and especially doppelbocks, these flavors are desirable. They can also often be found in barley wines. In other styles, however, they may be less desirable, or even a flaw.

One of the most effective methods of encouraging melanoidin production is a decoction mash. This will not only produce melanoidins in the mash, but will encourage further production in the boil. Other factors that increase melanoidin production are higher temperatures, a pH in the 5.0 - 5.4 range, and higher levels of amino acids. There is also a Melanoidin malt on the market for bootsing melanoidin flavor.

Increasing the concentration of the wort will promote carmelization, but Maillard reactions will occur whether the liquid is concentrated or dilute.

Mallard reactions, on the other hard, are usually a result of late-boil fowl additions...

Without fully answering your question, the Maillard Reaction is a chemical process that foods undergo as they're cooked. In general it's visibly seen as the browning of foods as they heat, and is responsible for generating rich and complex flavors. In cooking, the product of the reaction is nicknamed "GBD" or "Golden brown delicious".

In terms of beer, the reaction is important when barley is malted (the browning which occurs is partly due to the reaction), as well as during the boiling of the wort. During the boil the rim of the pan will develop a brownish sludge, which--if stirred into the wort--will impart it's flavor which can be significantly different than the raw wort flavor.

Again, as I'm not an expert, I won't try to advise whether you should or shouldn't encourage the reaction (I'm very curious for other input though)--but I can assure you the effect of the reaction plays a significant role in the flavor.

To add some further information, maillard reactions continue to happen even after the beer is bottled. Beers that are cellared will continue to darken and change in complexity due to the maillard reactions. They also can create properties that stale the beer. So there is a fine line in cellaring between adding complexity and adding stale characteristics. Since heat is a factor in the maillard reaction the temperature that you store your beer at affects how fast the beer will stale. You can thus extend the drinkability life of a beer by storing it at refrigerator temps.